No idea if this helps...
... but I remember some threads from years ago where the LPG genset didn't want to run on small tanks. It seemed (at least to me) that the genset ran on vapor and the problem was that the cooling effect you noticed. As the LPG is drawn off QUICKLY (genset uses a lot, relative to an appliance), it cools, and the cooling causes the pressure in the tank to drop to where it doesn't flow as intended.
At least that was the implication of what I read. I think the trailer owner tried a larger LPG tank and it helped.
All this tells me that if cooling of the LPG tank is a problem, the outdoor temperature you're operating in could make a difference. At 90* and you want to run A/C with the genset, your tank might work. At 30* and wanting to run heaters, might not.